[Insert Fig. 2 here]
2.3 Validation of the DRM on trees
2.3.1 Lysimeter experiments
We used whole-tree lysimetry to test the DRM, as well as the HRM, CHPM and Tmax methods. Measurements were made at the Centre for Carbon, Water and Food at the Camden Campus of the University of Sydney in Brownlow Hill, NSW, Australia (34.03oS, 150.66oE) in 2016-2017. Three E. cypellocarpasaplings, each in a large container (0.9 m by 1.2 m by 0.9 m), were placed on lysimeters capable of weighing up to 1200 kg (50 g resolution, Mettler Toledo). The three saplings were originally planted as seedlings in the containers in 2011. Irrigation was adjusted so that we could measure sap velocity across a range of water availabilities. Trees were initially irrigated 30 minutes (around 2.5 L min-1) twice a day for a month, and then reduced to 20 minutes twice a day, then to two minutes six times a day, and finally without any irrigation for several days. Transpiration was calculated from weight loss measured by the weighing lysimeters. Additional details regarding the lysimetry data analysis are provided in Methods S3 .
2.3.2 Sap velocity probe construction and installation
Each probe set comprises three temperature sensor probes and one heater probe. Each temperature sensor contained one thermistor (QTI Sensing Solutions, E320) inserted into an 18-gauge blunt tip needle. Thermistor sensing tips were coated in heat sink compound before inserting and then centered 1.5 cm from the distal end of the needle. Thermistor depths were fixed by placing a drop of low-viscosity cyanoacrylate glue in both the proximal and distal end of the needle. Each heater was made by feeding approximately 65 cm Manganin wire (Goodfellow Group Ltd., CU065822) through a 27.5-gauge hypodermic needle, leaving 3 to 5 cm of wire extending from the proximal end of the needle with the remainder (approximately 60 cm) protruding from the distal end, and then winding the wire tightly around the outside of the needle until 3 cm of needle was covered with winding. Cyanoacrylate glue was then applied to the proximal section of winding to prevent further unwinding, and the needle’s plastic base was removed by gripping the needle with pliers below the glued winding and bending it repeatedly at an angle of approximately 20 degrees until it fractured and could be pulled off the remaining wire. The resulting ”heater core” was then dipped in heat sink compound and inserted into an 18-gauge needle. The total resistance of the heaters varied from 17 to 18.4 Ohms and was insensitive to temperature between 0⁰C and 70⁰C. The sensors and heaters were soldered to extension cables which were wired to an AM16-32B multiplexer (Campbell Scientific Inc., Logan, Utah) and relays respectively. Initiation of the heat pulses and the temperature recordings were controlled by a CR850 datalogger (Campbell Scientific).
Probes were inserted radially into the sapwood and parallel to one another, with the aid of a portable drill press (Kanzawa, K-801) and a rigid levelling plate made from angle iron and strapped to each tree. Temperature sensors were located at distances of 0.75 cm proximal (Probe #1, x 1 = -0.75 cm), 0.75 cm distal (Probe #2,x 2 = +0.75 cm) and 2.25 cm distal (Probe #3,x 3 = +2.25 cm) to the heater probe. One set of sensors was installed on each tree approximately 30 cm above the soil surface. A heat pulse of 7-16 s in duration was initiated every 15 or 30 min by applying 12 V across the heater probe wire, which induced a peak temperature rise of around 2 K in Probe #2. The subsequent temperature change of each probe was recorded every second for 5 to 10 min. All probe installations were wrapped with 15 cm thick polyester insulation and covered with Mylar-coated bubble wrap for insulation.
The temperature rise (δ i) was calculated by first computing the initial temperature of each probe as its average for 5 s before a given heat pulse. To account for drift in background temperature during the rise and decay of temperature following each heat pulse, we used the pchip function in Matlab to simulate changes in background temperature by smooth interpolation between each successive pre-heat-pulse initial temperature, and then subtracted the resulting changes in background temperature from each time course of temperature to give δ i.
Heat pulse velocities were converted to sap velocity following Burgess et al. (2001). The sapwood properties includingρ b and mc were measured by tree core samples during and at the end of the experiments (see sapwood properties in Table S1). Whole-tree sap flow (kg h-1) was determined as the product of sap velocity (cm h-1 = cm3sapcm-2sapwood h-1), sapwood cross sectional area (cm2sapwood) and a conversion factor (10-3 kg cm-3sap). The diameters of the eucalypt trees (#1, 2 and 3) were 8.2, 8.4 and 10.0 cm, respectively, at the height of each heater probe installation. As we did not attempt to account for radial or azimuthal variation in sap velocity (Lopez-Bernal, Alcantara, Testi, & Villalobos, 2010; Phillips, Oren, & Zimmermann, 1996; Poyatos, Cermak, & Llorens, 2007), we focused on consistency between the dynamics of water loss as measured by lysimetry and that inferred by sap flow methodologies, rather than quantifying absolute sap velocity. We thus empirically adjusted whole-tree sap flow estimates using a scaling factor for each tree that equalized peak midday flow for lysimetry and sap flow methods. Scaling factors were close to constant from February through May 2016 (see Table S3). Comparisons of sap flow methods to lysimetry are best made when flows of water into or out of stores in the stem are least, typically during the middle of the day (greatest in early morning and late afternoon (Thomas N. Buckley et al., 2011; Chuang, Oren, Bertozzi, Phillips, & Katul, 2006; Deng et al., 2017).
3 Results
3.1 Theoretical testing
3.1.1 Operational procedures for DRM
We calculated V DRM by averaging values (measured at 1 Hz) for window sizes varying between 5 and 100 s, based on a central time point at which either σ 12 orσ 23 was smallest. Monte Carlo simulations indicated that the optimal window size was inversely related to sap velocity (Fig. 3 ). For example, standard errors were least using a window width of 80-100 s for a true sap velocity of 30 cm h-1, but 20-40 s when velocities were 80 cm h-1 (Fig. 3b,e ). Time windows for the least biased estimates of V did not overlap as indicated by the SE atV = 80 cm h-1 when comparing time windows of 20 and 80 s (Fig. 3e ,f ). This is due to the assumption of an instantaneous heat pulse in the Marshall Model, from which all heat-pulse based methods are derived (see Discussion).